Approach for differentiating nanoscale materials based on the size 

criterion using number of modes by count and size, locations of 

modes by size, mean size count, and standard deviations centered 

around mean sized count. 

Background

EPA is proposing in its proposed 8(a) and significant new use rules to differentiate discrete 

nanoscale materials (NMs) based on the numeric middle of a particle size range whose 

resolution varies according to the width of the range (the current proposal outlines five 

ranges: 1 – 20 nm; 20 – 50 nm; 50 – 1000 nm; and 1000 – 10000 nm). For example, the 

numeric middle of a NM with 90% of the primary particle sizes falling between 55 - 65 nm 

is 60 nm. There are several considerations and parameters to consider beyond this specific

approach:

1. No scientific literature has differentiated or identified NMs based on the numeric 

middle of the size range. 

2. Extensive scientific and mathematical literature has dealt with distributions of 

various properties and typically identifies four properties to characterize a 

distribution, as well as the type of distribution itself: mean or average, variance1, 

skewness2, and kurtosis3. Common distributions include: normal, Poisson, 

Rayleigh, exponential, and geometric, but there are many others. 

3. Figures 1-3 illustrate different particle size distributions that, under the proposed 

approach, would be considered to be the same “nanoscale material” despite 

scientific studies demonstrating that these substances have different environmental 

properties (Auffan, Rose et al. 2009):

a. Figure 1 

 

1 Measures the “width” of the distribution around the mean. For population samples, this is expressed as 

1

( )

2

2









N

X x

s

i

i

, where Xi is the data point and x is the sample mean. 

2 Skewness is a measure of the asymmetry of the probability distribution around the mean. A negative value 

indicates a longer tail on the left, while a positive number indicates a longer tail on the right. 

3 Kurtosis is a measure of the "peakedness" of the probability distribution. Higher kurtosis means more of the 

variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations.
Figure 1 illustrates a broad distribution and a narrow one that have the same “middle” of the range. The peaks 

are also constructed to have the same mean of particles by count, though not by mass. The black line indicates 

the mean at 50nm, which is also the middle of each range, while each dashed vertical line represents two

standard deviations from the mean (roughly 95% of the particle size distribution) for the respective 

distributions. 

Figure 2 illustrates two materials have different means and skewness (there is >5nm difference between the 

means), but 95% of the particle s fall into the same size range (represented by the dashed vertical lines) . The 

distribution for material 1 has relatively small particles and is skewed left, while the distribution for material 2 

has relatively large particles and is skewed right. Even though the majority of the particle sizes are different in 

each distribution, they would still be considered the same “material” under the current proposal. The mean by 

count for each material are represented by the solid vertical lines. 

0 10 20 30 40 50 60 70 80 90 100

Frequency

Particle size in nanometers

Material 1 -

broad 

distribution

Material 2 -

narrow 

distribution

0 10 20 30 40 50 60 70 80 90 100

Frequency

Particle size in nanometers

Material 1 -

skewed left

Material 2 -

skewed right
Figure 3 illustrates a material with bimodal distribution (#1) with the same mean and 

#2. Line a represents the mean for both substances, which also happens to be the median for material #2. The 

median for material #1 is at line b. Lines c & d are the modes for material #1. 

Although the proposed approach as d

understanding of and compliance with the rule

parameters based on different size distributions su

approach. At the same time, more complexity o

would be unable to meet the requirements, because the proposed revised approach is 

based on statistical distributions science. 

Foundations for an alternative

Although the mathematics of distributions 

differentiation of NMs based on

distribution or distributions, such as utilizing Student’s T

There are several reasons for th

1. There are several different types of “particle” including, but not limited to:

a. Primary particle

b. Grain

c. Aggregate

d. Agglomerate

No single method or protocol exists for characterizing these properties of a single 

material that can span orders of magnitude of size or to adjust for different 

conditions. 

0 20

Frequency

illustrates a material with bimodal distribution (#1) with the same mean and width of range as material 

#2. Line a represents the mean for both substances, which also happens to be the median for material #2. The 

median for material #1 is at line b. Lines c & d are the modes for material #1. 

approach as described in the current rule could simplify 

understanding of and compliance with the rule, the differences in environmental 

parameters based on different size distributions support the idea for a more rigorous 

approach. At the same time, more complexity or nuance does not mean a regulated entity 

would be unable to meet the requirements, because the proposed revised approach is 

utions science. 

n alternative approach

Although the mathematics of distributions has been extensively described, a rigorous 

n particle size distributions utilizing a particular type of 

, such as utilizing Student’s T-test, is probably not warranted. 

There are several reasons for this:

There are several different types of “particle” including, but not limited to:

No single method or protocol exists for characterizing these properties of a single 

material that can span orders of magnitude of size or to adjust for different 

40 60 80

Material 1 

material

Material 2 

material

d

b c

a

width of range as material 

#2. Line a represents the mean for both substances, which also happens to be the median for material #2. The 

simplify 

, the differences in environmental 

for a more rigorous 

r nuance does not mean a regulated entity 

would be unable to meet the requirements, because the proposed revised approach is 

been extensively described, a rigorous 

particle size distributions utilizing a particular type of 

is probably not warranted. 

There are several different types of “particle” including, but not limited to:

No single method or protocol exists for characterizing these properties of a single 

material that can span orders of magnitude of size or to adjust for different 

100

Material 1 - bimodal 

Material 2 - broad 
2. Under the Nanoscale Material Stewardship Program and the nanoscale notices 

received under the New Chemicals Program, the Agency has frequently seen particle 

size distributions that are bimodal size distributions typically representing 

aggregation and agglomeration processes. There are several other processes that 

might generate a bi- or trimodal distribution including particle formation processes 

that occur under threshold reaction conditions (this might also lead to graininess). 

There may be others.

3. Different shapes will also lead to issues of defining the boundaries of distributions of 

critical parameters for non-spheroid nanoscale materials like fibers, trees, etc. 

4. Utilizing statistical tools, such as a Student’s t-test, to identify whether a particular 

size distribution is the same or different than another size distribution is a 

scientifically objective approach to settling the question as to whether two materials 

are the same from a size perspective within specified limits of statistical uncertainty. 

However, this would require embedding the distribution information into the 

nomenclature of a discrete material. This strict approach is well outside the norm of 

how EPA considers distributions in Unknown Variable Composition and Biological 

(UVCB) substances for chemical nomenclature. Second, it would place a large 

burden on industry to submit and EPA to assess potentially new submissions. 

Taking into account the above issues, and given the need to balance simplicity and taking a 

case-by-case approach, the following framework is an alternative proposal for 

differentiating NMs from each other based on size. The other parameters outlined in the 

proposed rule would remain the same. 

Alternative approach

1. The key parameters of size distributions are:

a. Number of modes by count and size

b. Locations of modes by size

c. Mean size by count

d. Two standard deviations centered around mean size by count

2. Each chemical substance that meets the definition of a nanoscale material should 

be evaluated as to the number of type of critical dimensions. 

Scale of 

dimensions 

(a, b, and c)

Example 

Shapes

Number of 

critical 

dimensions

Dimensions Notes

abc Spheroid, 

cubes

1 Diameter Zingg’s shape: 

equant

abc Irregular 

shapes, 

cones

1 Equivalent 

spherical 

diameter

Submitter 

should convert 

volume or 

surface area of 
NM into an 

equivalent 

spherical 

diameter

ab<c Fiber, 

needle, 

rod, 

2 Length and 

diameter

ab should be 

converted to 

equivalent 

circular 

diameter

Zingg’s shape: 

prolate

ab>c Ring, 

sheet, 

torus

2 Depth and 

diameter

ab should be 

converted to 

equivalent 

circular 

diameter

Zingg’s shape: 

oblate

a<b<c Ribbons, 

attached 

or 

unattached

3 Length, width, 

depth

Zingg’s shape: 

bladed

Two dimensions should be considered equivalent or on the “same scale” when each 

dimension’s mean by count are within 3/2 of each other. This approach was 

developed by Zingg for use in describing 3-dimensional objects in geology (Zingg 

1935)}. Materials that are “hollow” or that encapsulate another substance should 

have another critical dimension: internal volume. 

3. Each critical dimension should be defined by the four parameters of the size 

distribution as follows:

a. Number of modes by integer number provided the distribution is 

statistically relevant. Hence, a bimodal distribution is different than 

unimodal distribution. A mixture of two normal distributions with equal 

standard deviations is bimodal only if their means differ by at least twice 

the common standard deviation (Schilling, Watkins et al. 2002)

b. Locations of modes by size, mean size by count, two standard deviations 

centered around mean size by count by the following approaches to size 

resolution:

i. for parameters up to 20 nm, the parameter differs from that of other 

materials by 2 nm or more,

ii. for parameters between 20 and 50 nm, the parameter differs from 

that of other materials by 5 nm or more,

iii. for parameters between 50 and 1000 nm, the parameter differs from 

that of other materials by 10 nm or more,
iv. for parameters between 1000 and 10000 nm, the parameter differs 

from that of other materials by 100 nm or more,

v. for parameters greater than 10000 nm, the parameter differs from 

that of other materials by 1000 nm or more

4. Nanoscale materials shall be distinguished by distributions at the primary, 

secondary, and tertiary structural level, e.g. primary particles, aggregates, and 

agglomerates. Hence, an aggregate with a mean size by count of 8 microns would 

be different than an aggregate with a mean size by count of 4 microns, even if the 

nanoscale primary particles of each substance would be considered to have the 

same size distribution. 

Auffan, M., J. Rose, et al. (2009). "Towards a definition of inorganic nanoparticles from an 

environmental, health and safety perspective." Nat Nano advance online 

publication.

Schilling, M. F., A. E. Watkins, et al. (2002). "Is Human Height Bimodal?" The American 

Statistician 56(3): 223-229.

Zingg, T. (1935). "Beitrag zur Schotteranalyse." Schweizerische Mineralogische und 

Petrographische Mitteilungen 15: 39–140.